L11a355

(Knotscape image)

See the full Thistlethwaite Link Table (up to 11 crossings).

Planar diagram presentation	X_12,1,13,2 X_18,7,19,8 X_14,3,15,4 X_16,5,17,6 X_20,10,21,9 X_22,19,11,20 X_4,15,5,16 X_6,17,7,18 X_8,22,9,21 X_2,11,3,12 X_10,13,1,14
Gauss code	{1, -10, 3, -7, 4, -8, 2, -9, 5, -11}, {10, -1, 11, -3, 7, -4, 8, -2, 6, -5, 9, -6}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 3 u 4 + v 2 u 4 - v 4 u 3 + 3 v 3 u 3 -2 v 2 u 3 + v u 3 + v 4 u 2 -2 v 3 u 2 + 3 v 2 u 2 -2 v u 2 + u 2 + v 3 u -2 v 2 u + 3 v u - u + v 2 - v$ (db)
Jones polynomial	$-\frac{1}{\sqrt{q}}+\frac{2}{q^{3/2}}-\frac{4}{q^{5/2}}+\frac{5}{q^{7/2}}-\frac{7}{q^{9/2}}+\frac{8}{q^{11/2}}-\frac{8}{q^{13/2}}+\frac{7}{q^{15/2}}-\frac{6}{q^{17/2}}+\frac{3}{q^{19/2}}-\frac{2}{q^{21/2}}+\frac{1}{q^{23/2}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$- z 5 a 9 -4 z 3 a 9 -3 z a 9 + z 7 a 7 + 5 z 5 a 7 + 8 z 3 a 7 + 6 z a 7 + a 7 z -1 + z 7 a 5 + 4 z 5 a 5 + 2 z 3 a 5 -3 z a 5 - a 5 z -1 - z 5 a 3 -4 z 3 a 3 -3 z a 3$ (db)
Kauffman polynomial	$- z 4 a 14 + 2 z 2 a 14 -2 z 5 a 13 + 4 z 3 a 13 -2 z a 13 -2 z 6 a 12 + 2 z 4 a 12 -2 z 7 a 11 + 2 z 5 a 11 -3 z a 11 -2 z 8 a 10 + 4 z 6 a 10 -5 z 4 a 10 + z 2 a 10 -2 z 9 a 9 + 7 z 7 a 9 -13 z 5 a 9 + 10 z 3 a 9 -2 z a 9 - z 10 a 8 + 2 z 8 a 8 -3 z 4 a 8 + 4 z 2 a 8 -4 z 9 a 7 + 18 z 7 a 7 -30 z 5 a 7 + 26 z 3 a 7 -9 z a 7 + a 7 z -1 - z 10 a 6 + 2 z 8 a 6 + 3 z 6 a 6 -5 z 4 a 6 + 3 z 2 a 6 - a 6 -2 z 9 a 5 + 8 z 7 a 5 -8 z 5 a 5 + 5 z 3 a 5 -5 z a 5 + a 5 z -1 -2 z 8 a 4 + 9 z 6 a 4 -10 z 4 a 4 + 2 z 2 a 4 - z 7 a 3 + 5 z 5 a 3 -7 z 3 a 3 + 3 z a 3$ (db)

The coefficients of the monomials

t r q j

are shown, along with their alternating sums

χ

(fixed

j

, alternation over

r

). The squares with yellow highlighting are those on the "critical diagonals", where

j -2 r = s + 1

j -2 r = s -1

, where

s =

-5 is the signature of L11a355. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -6$	$i = -4$
$r = -9$	${\mathbb Z}$
$r = -8$	${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -7$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -6$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{3}$
$r = -5$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -4$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = -3$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = -2$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = -1$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 0$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{3}$
$r = 1$	${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = 2$	${\mathbb Z}_2$	${\mathbb Z}$